How do I move a vertex along the normal axis via scripting?
if you only change v.co.z all will follow the z axis... 

Try :

for v in bm.verts:
    r = random.uniform(-0.05,0.05)
    v.co.x += v.normal.x * r
    v.co.y += v.normal.y * r
    v.co.z += v.normal.z * r


As batFINGER says in the comment, that can be written as :

for v in bm.verts:
    r = random.uniform(-0.05,0.05)
    v.co += r * v.normal